What is (−8) squared?

It is 64, not −64. Squaring means multiplying the number by itself, and (−8)² = (−8) × (−8). A negative times a negative is positive, so the two minus signs cancel and the answer is +64. Leaving it as −64 keeps a single minus sign, but squaring a negative always gives a positive result.

Is −4² the same as (−4)²?

No. (−4)² = 16, but −4² = −16. The bracket decides what gets squared. In (−4)² the whole of −4 is squared, so (−4) × (−4) = 16. In −4² there is no bracket, so by order of operations the square attaches to the 4 only: it means −(4²) = −(16) = −16. The minus sits outside the squaring.

Why does the bracket change the answer when squaring a negative?

Because powers are worked out before the minus sign is applied, unless a bracket forces the minus inside first. Without a bracket, −4² is read as −(4²): square the 4 to get 16, then apply the minus for −16. With a bracket, (−4)² makes −4 a single value that is squared, so (−4) × (−4) = 16. Same digits, opposite signs — the bracket is the whole difference.

GCSE Maths Foundation · AQA · Order of operations & negatives

Squaring a negative: (−8)² = 64, and −4² is not (−4)²

Squaring a negative trips students two ways. First, they keep the minus, writing $(-8)^2 = -64$ when squaring multiplies the number by itself and a negative times a negative is positive, so $(-8)^2 = 64$ . Second, they read $-4^2$ and $(-4)^2$ as the same, when the bracket decides: $(-4)^2 = 16$ but $-4^2 = -16$ .

The thirty-second fix: squaring a negative always gives a positive, because the two minus signs cancel — and the bracket decides what gets squared, so −4² means −(4²) = −16, while (−4)² squares the whole −4 to give 16. So $(-8)^2 = 64$ , and the killer pair is $-4^2 = -16$ versus $(-4)^2 = 16$ .

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How to spot it in your own work

You kept the minus on a square, writing $(-8)^2 = -64$ instead of 64 — a negative times a negative is positive.
You read $-4^2$ and $(-4)^2$ as the same, when they are $-16$ and $16$ .
You squared the whole of $-4^2$ as if it were bracketed, getting 16 instead of $-16$ — without a bracket the square reaches only the 4.
You substituted a negative into $x^2$ and left the answer negative, e.g. taking $(-3)^2$ as $-9$ rather than $9$ .

An exam question that triggers it

Here is a canonical AQA Foundation trigger (non-calculator paper, work out the value):

Work out

$(-8)^2$

The misconception is to carry the minus through and write $-64$ , as if squaring kept the sign. But squaring means $(-8) \times (-8)$ , and a negative times a negative is positive.

The two minus signs cancel: $(-8) \times (-8) = 64$ . Squaring any negative gives a positive result.

Why students fall for this

A minus sign feels like it should travel with the number, so squaring it looks like it ought to keep it — hence $-64$ . But squaring is multiplying the number by itself, and $(-8)^2 = (-8) \times (-8)$ . Two negatives multiplied give a positive, so the minus signs cancel and the answer is $+64$ . Every squared negative comes out positive for the same reason.

The bracket version is sharper. $-4^2$ and $(-4)^2$ look almost identical but are not equal, because powers are worked out before the minus is applied unless a bracket forces the minus inside first. Without a bracket, $-4^2$ means $-(4^2) = -16$ : square the 4, then apply the minus. With a bracket, $(-4)^2$ makes $-4$ a single value that is squared, so $(-4) \times (-4) = 16$ .

AQA Foundation papers exploit both directly: evaluating a bracketed square like $(-8)^2$ , and substituting a negative into an expression with $x^2$ where the bracket — or its absence — fixes the sign.

Worked example — the killer pair. Work out $-4^2$ and $(-4)^2$ .

No bracket means the square reaches only the 4; a bracket means the whole $-4$ is squared:

-4^2 = -(4^2) = -16 \qquad (-4)^2 = (-4)\times(-4) = 16

The trap is to treat them as the same. They differ only by the bracket, and the bracket is the whole point: $-16$ against $16$ .

The fix: Squaring a negative gives a positive; the bracket decides what gets squared

Squaring multiplies the number by itself. $(-8)^2 = (-8) \times (-8)$ , and a negative times a negative is positive, so the answer is $64$ — the minus signs cancel.

A squared negative is always positive. Whatever the digits, $(-n)^2$ comes out positive, because you are multiplying two negatives.

Look for the bracket. With a bracket, the whole negative is squared: $(-4)^2 = 16$ . Without one, the square attaches to the digit only and the minus stays outside: $-4^2 = -(4^2) = -16$ .

When substituting, bracket the negative. Putting $x = -3$ into $x^2$ means $(-3)^2 = 9$ , not $-9$ — write the bracket so the square reaches the whole value.

Worked example

Work out $(-8)^2$ , then compare $-4^2$ with $(-4)^2$ . These are the two traps: keeping the minus on a square, and missing the bracket.

Write the square as a product. $(-8)^2 = (-8) \times (-8)$ .
Cancel the signs. A negative times a negative is positive, so
$(-8)^2 = (-8)\times(-8) = 64$
The trap answer $-64$ keeps a single minus that should have cancelled.
Square without the bracket. $-4^2$ has no bracket, so the square reaches only the 4: $-(4^2) = -(16) = -16$ .
Square with the bracket.
$(-4)^2 = (-4)\times(-4) = 16$
Same digits, opposite signs: $-4^2 = -16$ but $(-4)^2 = 16$ . The bracket is the whole difference.

So a squared negative is positive whenever the negative is bracketed, and the opening trigger is $(-8)^2 = 64$ — not the $-64$ you get by carrying the minus through the square.

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Common questions

What is $(-8)^2$ ?: It is 64, not $-64$ . Squaring means multiplying the number by itself, and $(-8)^2 = (-8) \times (-8)$ . A negative times a negative is positive, so the two minus signs cancel and the answer is $+64$ . Leaving it as $-64$ keeps a single minus sign, but squaring a negative always gives a positive result.
Is $-4^2$ the same as $(-4)^2$ ?: No. $(-4)^2 = 16$ , but $-4^2 = -16$ . The bracket decides what gets squared. In $(-4)^2$ the whole of $-4$ is squared, so $(-4) \times (-4) = 16$ . In $-4^2$ there is no bracket, so by order of operations the square attaches to the 4 only: it means $-(4^2) = -(16) = -16$ . The minus sits outside the squaring.
Why does the bracket change the answer when squaring a negative?: Because powers are worked out before the minus sign is applied, unless a bracket forces the minus inside first. Without a bracket, $-4^2$ is read as $-(4^2)$ : square the 4 to get 16, then apply the minus for $-16$ . With a bracket, $(-4)^2$ makes $-4$ a single value that is squared, so $(-4) \times (-4) = 16$ . Same digits, opposite signs — the bracket is the whole difference.

Related misconceptions

Ordering and signs of negative numbersThe underlying sign rules: a negative times a positive is negative and further left is smaller, so −3 × 3 = −9 and −7 < −5.
Order of operations: × and ÷ before + and −Why the bracket matters at all: powers and brackets come before the rest, so −4² means −(4²) while (−4)² squares the whole value.

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